Abstract

The author has studied the existence of periodic solutions of a type
of higher order delay functional differential equations with neutral type by using the theory of
coincidence degree, and some new sufficient conditions for the existence of periodic solutions have
been obtained.

1. Introduction and Lemma

With the rapid development of modern science and technology, functional differential equation with time delay has been widely applied in many areas such as bioengineering, systems analysis, and dynamics. Functional differential equation with complex deviating argument is an important type of the above function. Because the property of the solution to this kind of equation is impossibly estimated, so the literature on the functional differential equation with complex argument is relatively rare [1]. In recent years, with the maturity of the theory of nonlinear functional analysis and algebraic topology, we have the powerful tools of the study on the functional differential equation with complex deviating argument, so it is possible to study the above equation. Furthermore, the study on the periodic solutions of functional differential equation is always one of the most important subject that people concerned for its widespread use. Many results of the study of Duffing-typed functional differential equation and Liénard-typed functional differential equation have been obtained, for example, the literatures [2–18]. Hitherto, the literature of the discussion of higher order functional differential equations has not been found a lot [19]. In this paper I have studied and derived some sufficient conditions that guarantee the existence of periodic solutions for a type of higher order functional differential equations with complex deviating argument as the following:
and some new results have been obtained.

In order to establish the existence of -periodic solutions of , we make some preparations.

Definition 1.1. Let , are Banach spaces, and let be an open and bounded subset in , and let be linear mapping; the mapping will be called a Fredholm mapping of index zero if and is closed in .

Definition 1.2. Let , let be projectors, and let be nonlinear mapping; the mapping will be called -compact on if and are compact.

Lemma 1.3 (see [20]). Let , be Banach spaces; is a Fredholm mapping of index zero ; are continuous mapping projectors; is an open bounded set in ; is -Compact on , furthermore suppose that:(a);
(b);
(c),
then the equation has at least one solution on , where is Brouwer degree.

2. Main Results and Proof of Theorems

Theorem 2.1. Suppose that , , , are continuous for their variables, respectively, , , , and furthermore suppose that(a), when , such that ;(b), such that ;(c),
where , and , then has at least one -periodic solution.

Proof of Theorem 2.1. In order to use continuation theorem to obtain -periodic solution of , we firstly make some required preparations. Let
and the norm of and is , , , and , respectively; then the and with this norm are Banach spaces.Firstly, we study the priori bound of -periodic solution of following equation:
Suppose that is an arbitrary -periodic solution of (2.2), put into, (2.2) and then integrate both sides of (2.2) on , so
For the continuity of , , , there must exist a number such that
that is,
For the condition (a) of Theorem 2.1, we have
Let
so
In view of
we have
that is,
Noting , so there must exist the number such that , where .For all,
we have
that is,
Combining (2.11), (2.14), we get
By (2.2), we get
where , , and .Noting (2.14) and the conditions (b), (c) of Theorem 2.1, we have
so
where .Let
that is,
Noting (2.14), (2.15), and (2.20), we have
Let , and let ; then is an open and bounded set in .
Let
then the corresponding equation of is (2.2).Now, we define projection operators as follows;
Obviously, , are continuous operators, , , and it is easy to prove that is a Fredholm mapping of index zero and is -Compact on .From the above discussion and the construction of , we know that for all , , , therefore the condition (a) of Lemma 1.3 holds.For arbitrary , , by the definition of , , we have
so
therefore the condition (b) of Lemma 1.3 holds.Making a transformation.
we have
So , that is, is a homotopy, = , where is an identity mapping, and the condition (c) of Lemma 1.3 holds.From above all, the requirements of Lemma 1.3 are all satisfied, so has at least one -periodic solution under the condition of Theorem 2.1, so the proof of Theorem 2.1 is completed.

Remark 2.2. In Theorem 2.1, if and the condition (a) of Theorem 2.1 is when , , and the rest are unchangeable, then has at least one -periodic solution.

If the is not a bounded function, we have the following theorem.

Theorem 2.3. Suppose that , , , are continuous for their variables, respectively, , , , and furthermore suppose following:(a), when , such that ;(b), such that ;(c),
where , and , then has at least one -periodic solution.

Proof of Theorem 2.3. Banach spaces , and the mappings , , , and are the same to Theorem 2.1, and their property are equal to Theorem 2.1, then the corresponding equation of is
It is similar to Theorem 2.1, there must exist a number , such that
and it is easy to obtain
Noting (2.28), (2.30) and the conditions (b), (c) of Theorem 2.3, we have
So
where , and .Let
that is,
Noting (2.30) and (2.34), we have
Let , and we take ; then is an open and bounded set in .Similarly to Theorem 2.1, we prove easily that is a Fredholm mapping of index zero and is -compact on and the conditions (a), (b), and (c) of Lemma 1.3 hold.From above all, the requirements of Lemma 1.3 are all satisfied, so has at least one -periodic solution under the condition of Theorem 2.3, so far the proof of Theorem 2.3 is completed.

Remark 2.4. In Theorem 2.3, if and the condition (a) of Theorem 2.3 is when , , and the rest are unchangeable, then has at least one -periodic solution.

If the , we have the following theorem.

Theorem 2.5. Suppose that , , , are continuous for their variables, respectively, , and meet the condition of Theorem 2.1 and furthermore suppose as follows:(a);(b), such that ;(c),
where , and , then has at least one -periodic solution.

Proof of Theorem 2.5. Banach spaces , and the mappings , , , and are the same to Theorem 2.1, and their property are equal to Theorem 2.1, then the corresponding equation of is
Suppose that is an arbitrary -periodic solution of (2.36), put into (2.36), and then integrate both sides of (2.36) on , so
For the continuity of , , , there must exist a number , such that
Combing the condition (a) of Theorem 2.5, there must exist , such that
Similarly to Theorem 2.1, we have
By (2.36), (2.37), (2.39), and (2.41) and the conditions (b), (c) of Theorem 2.5, we have
So
Let
that is,
Noting (2.40), (2.41), and (2.45), we have
For condition (a), there exist and 0, such that ; let , and we take ; then is an open and bounded set in .Similarly to Theorem 2.1, we prove easily that is a Fredholm mapping of index zero and is -compact on and the conditions (a), (b), and (c) of Lemma 1.3 hold.From above all, the requirements of Lemma 1.3 are all satisfied, so has at least one -periodic solution under the condition of Theorem 2.5, so the proof of Theorem 2.5 is completed.

Remark 2.6. In Theorem 2.5, if and the condition (a) of Theorem 2.1 is when , , and the rest are unchangeable, then has at least one -periodic solution.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (11101305).